Group theory and generalizations Representation theory of groups Theorem

Schur orthogonality relations

Builds on Group Group representation Irreducible representation Character of a representation Conjugacy class Centraliser Unitary matrix Complex conjugate

Statement

Let $G$ be a finite group. For functions $\alpha, \beta \colon G \to \C$ write $\ip{\alpha}{\beta} = \frac{1}{\abs{G}} \sum_{g \in G} \alpha(g) \overline{\beta(g)},$ where the bar is the complex conjugate.

Let $\rho$ and $\sigma$ be irreducible complex representations of $G$ by unitary matrices, of degrees $d_\rho$ and $d_\sigma$ . If $\rho$ and $\sigma$ are not isomorphic, then $\ip{\rho_{ij}}{\sigma_{kl}} = 0$ for all indices. For all indices $i, j, k, l$ one has $\ip{\rho_{ij}}{\rho_{kl}} = \delta_{ik}\delta_{jl} / d_\rho$ .
Let $\chi$ and $\psi$ be irreducible complex characters of $G$ . Then $\ip{\chi}{\psi} = 1$ if $\chi = \psi$ and $\ip{\chi}{\psi} = 0$ otherwise.
Let $g, h \in G$ , and let $C_G(g)$ be the centraliser of $g$ . Then $\sum_{\chi} \chi(g) \overline{\chi(h)}$ , summed over all irreducible complex characters $\chi$ of $G$ , equals $\abs{C_G(g)}$ if $g$ and $h$ are conjugate, and $0$ otherwise.

Remarks

The proof of the first relation applies Schur's lemma to the averaged map $\sum_{g} \rho(g) A \sigma(g)^{-1}$ , which commutes with the action for every matrix $A$ . Taking traces in the first relation gives the second. The second relation shows that the irreducible characters are orthonormal in the space of class functions; they span this space, and the third relation follows from that. For example, for $G = \Z/2\Z$ the two irreducible characters are $(1, 1)$ and $(1, -1)$ , and $\ip{\chi}{\psi} = \tfrac{1}{2}(1 - 1) = 0$ . A consequence: a complex representation with character $\chi$ is irreducible if and only if $\ip{\chi}{\chi} = 1$ . Every complex representation of a finite group is isomorphic to a unitary one, so the first relation applies to all irreducible representations after a change of basis.

Going deeper

For a student

The relations are the main computational tool of character tables. Row orthogonality (relation 2) gives the multiplicity $\ip{\chi_V}{\chi}$ of an irreducible $\chi$ in a representation $V$ . Column orthogonality (relation 3) with $g = h = 1$ gives $\sum_\chi \chi(1)^2 = \abs{G}$ . Together they show that the number of irreducible characters equals the number of conjugacy classes, and they let one fill in missing entries of a character table. They are the finite group case of the Peter–Weyl theory for compact groups, where the average becomes an integral against Haar measure.

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Tag	`AY0B20QC04`
URI	`m2://AY0B20QC04`
Kind	statement, theorem
Writing	W0 stub
MSC	20C15
Status	This copy of the site has no build data. Run `m2 status AY0B20QC04`.
About	Character of a representation, Irreducible representation
Uses	Group, Group representation, Irreducible representation, Character of a representation, Conjugacy class, Centraliser, Unitary matrix, Complex conjugate
Same as	Wikidata Q7433034 Wikipedia Schur orthogonality relations
Changed	2 November 2026

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